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Problem Statement

6(a) Show ${\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}{\mbox{ if }}S(0)=0$.

6(b) If $S(0)\neq 0$ can you find ${\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]$ in terms of $\displaystyle S(0)$?

Answer

a)

Remember that dummy variable $\lambda \!$ was used in substitution such that $\lambda =t-t_{0}\!$

This means $s(\lambda )=s(t-t_{0})={\mathcal {F}}\left[S(f)-S(f_{0})\right]\!$ In my notations Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(f_0)=S(f)|_{f=0 \!}

The problem statement says let $f_{0}=0\!$ where $S(0)=S(f)|_{f=0}=\int _{-\infty }^{\infty }s(t)e^{-j2\pi ft}dt=\int _{-\infty }^{\infty }s(t)dt\!$

${\mbox{ if }}S(0)=0\,\,\,\int _{-\infty }^{\infty }s(t)dt=0\!$

</math> and $\int _{-\infty }^{t}s(\lambda )\,d\lambda =\int _{-\infty }^{t}{\mathcal {F}}\left[S(f)-S(f_{0})\right]\,d\lambda \!$

$\int _{-\infty }^{t}s(\lambda )\,d\lambda =\int _{-\infty }^{t}{\mathcal {F}}\left[S(f)\right]\,dt\!$

${\mathcal {F}}^{-1}\left[S(f)-S(f_{0})\right]=\int _{-\infty }^{t}e^{j2\pi ft}\,dt\int _{-\infty }^{\infty }S(f)\,df={\frac {e^{j2\pi ft}}{j2\pi f}}\int _{-\infty }^{\infty }S(f)\,df=\!$

$\int _{-\infty }^{t}s(\lambda )\,d\lambda =\int _{\infty }^{\infty }S(f){\frac {e^{j2\pi ft}}{j2\pi f}}\,df={\mathcal {F}}^{-1}\left[{\frac {S(f)}{j2\pi f}}\right]\!$

Therefore ${\mathcal {F}}\left[\int _{-\infty }^{t}s(\lambda )\,d\lambda \right]={\frac {S(f)}{j2\pi f}}\!$